Solid State Theory. (Theorie der kondensierten Materie) Wintersemester 2015/16

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1 Martin Eckstein Solid State Theory (Theorie der kondensierten Materie) Wintersemester 2015/16 Max-Planck Research Departement for Structural Dynamics, CFEL Desy, Bldg. 99, Rm Luruper Chaussee Hamburg, Germany Tel: +49 (0) Lecture Website: Literature: G. Czychol, Theoretische Festkörperphysik, Springer J. M. Ziman, Principles of the Theory of Solids, Cambridge University Press, N. W. Ashcroft and M. D. Mermin, Solid State Physics. A. Altland and B. Simons, Condensed Matter Field Theory, Cambridge University Press, Topics: Electrons in the periodic crystal (band-structure calculations); Interacting electrons: Screening, Landau s Theory of Fermi Liquids; Phonons; Superconductivity; Magnetism; 1

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3 Introduction In solid state physics, we aim to understand structure of solids phases of condensed matter systems, like metallic or insulating behavior, magnetism, superconductivity,... elementary excitations ( spectroscopical measurements, transport properties) For the understanding of many phenomena, quantum mechanics is essential. Examples: chemical binding and stability of matter explained by quantum effect. Our non-relativistic theory of everything would be given by the Hamiltonian H = p 2 i 2m + e 2 r i i<j i r j electrons + i P 2 i e 2 Z i Z j R i R j 2m + i<j nuclei e 2 Z k r i R k i,k e-n attraction (1) The typical binding energy of en electron to a nucleus is of the order of several electron volts (H-atom: 13.6eV =1Ry). With the basic energy-temperature conversion 1eV = k B Kelvin 300K 30meV 1Ry, (2) one can see that thermal fluctuations k B T are negligible compared to atomic energies at room temperature. The Hamiltonian (1) is thus useful for the understanding of properties of matter at K (this would be a plasma), but it is usually not a good starting point to learn about a solid at room temperature. Not even fundamental phenomena such as the formation of a crystal can be predicted fully abinitio, although it should be contained in the formulation. The Hamiltonian can be vied as the high-energy theory for condensed matter physics, from which low-energy effective theories for condensed matter phases emerge. The situation in low energy physics is opposite to high-energy physics, where one tries to uncover the high-energy theories from which the physical laws for atoms and nuclei emerge: 3

4 10 mev condensed matter: effective theories 10 ev atoms and nuclei MeV high-energy physics new ground states with new types of excitations (quasiparticles) and effective interactions: this lecture Examples for effective theories for the solid: Harmonic crystal Spin models ( magnetism of localized moments) Electronic quasiparticles 4

5 Periodic structures Most solids have crystalline order on the atomic scale (although, macroscopically they can form crystallites on the µm scale.) Exceptions are glasses (amorphous structures), liquid crystals, etc., see Fig. 1. Crystalline symmetries are essential for a theoretical description of solids, hence we will deal almost exclusively with crystalline structures in this lecture. Figure 1: Left: Translational, but no orientational symmetry (e.g., Spin glass). Middle: Orientational, but no translational symmetry (e.g., nematic order). Right: Perfect crystalline order. Classification of crystalline structures Bravais lattice: Infinite set of points generated via translations of one point by integer linear combinations of a set of given vectors a i { R} = {n 1 a 1 + n 2 a 2 + n 3 a 3 n i Z}, (3) a i primitive lattice vectors, linearly independent. Lattice looks identical from every of its points. Example: The vectors a 0 0 a 1 = 0,a 2 = a,a 3 = a generate the perfect cubic lattice with lattice spacing a. 5

6 Note: The choice of the primitive lattice vectors is not unique: If a i is a set of primitive vectors, a i generates the same lattice iff a i = j M ija i with an integer matrix M of determinant ±1. Primitive unit cell: The volume spanned by the primitive unit vectors is called the primitive unit cell. It has volume V = a 1 (a 2 a 3 ). (4) In general, a unit cell can be any volume that covers full space without overlaps when it is translated by every vector of the Bravais lattice. A special (uniquely defined) choice is the Wigner Seitz cell, which is chosen to be the set of all points closer to a point R of the Bravais lattice than any other point (see Fig. 2) Figure 2: Left: A primitive unit cell of the square lattice (which has not the full point group symmetry of the lattice) Right: The Wigner-Seitz cell. The Wigner-Seitz cell around a lattice point R is geometrically constructed by drawing the line from R to any other point of the lattice, and the plane perpendicular to the line half-way between the points. All these planes cut out the Wigner Seitz cell. Crystal Structure: A Bravais lattice is only the set of points generated by the translation operations. The full crystal structure is obtained by placing objects of typically lower symmetry (atoms, molecules) in the lattice. The objects within one unit cell is called the basis. Example: The honey-comb lattice is not a Bravais lattice, but a Bravais lattice with a basis of two atoms (Fig. 3) Figure 3: The honey-comb lattice. 6

7 Crystal Symmetries: A crustal symmetry is a linear transformation of space r r = Dr + a (5) which maps the structure onto itself. a : Translation. D : orthogonal transformation (rotation, reflection, inversion). All operations D define the point group of the crystal (D is a transformation which leaves one point of the lattice invariant). All symmetry operations (point group + translations) define the space group of the crystal. Note: The operations D do not necessarily form a sub-group of the space group; there can be rotations/reflections that are symmetries of the crystal only in combination with a translation: screw axis: rotation + translation along rotation axis glide plane: reflection + translation along reflection plane a 2 a 1 dashed line: glide plane (reflection + translation by a1/2) How many crystal symmetries are there? Important restriction: only 2,3,4,6 fold rotation axis is compatible with translational symmetry (Algebraic) proof: Consider rotation matrix around z: cos(ϕ) sin(ϕ) 0 D z (ϕ) = 0sin(ϕ) cos(ϕ) Trace of this matrix is invariant under transformation into a different basis, because Tr[U 1 DU] = TrD for any invertible matrix U. Transforming to the basis defined by the primitive vectors (note 7

8 that the matrix U above need not be orthogononal), the rotation must be a matrix with integer entries (it maps primitve vectors on primitive vectors). Hence D z must satisfy Tr D z (ϕ) =2cos(ϕ)+1! Z. The only solutions to this equation are ϕ =0, π 3, π 2, 2π 3,π. Note: There are structureswith apparent 5-fold coordination in reziprocal space (Bragg-peaks in X-ray diffraction, see below). These quasi-crystals are not periodic in space, but may be obtained by projecting a periodic structure in higher dimensional space onto a subspace along a direction not commensurate with the primitive vectors. The crystrallographic restriction finally restricts the number of Bravais lattices and crystal structures (i.e., Bravais lattice + basis): Bravais lattice crystal structure number of point groups 7 (4) 32 (13) number of space groups 14 (5) 230 (17) The number in brackets corresponds to spacial dimension d =2. The space groups in d =2are called wallpaper groups. The correspond to a symmetries of mosaic tilings of the plane. There are many good illustrations in the internet, e.g., https://de.wikipedia.org/wiki/ebene kristallographische Gruppe For a more detailed discussion of symmetries and Bravais lattices in 3d, see Ashcroft & Mermin, Chapter 6. Importance of symmetry for the description of solids: Quantum numbers, degeneracies in the spectrum (see band structure lectures). Symmetry determines the response coefficients of a solid. In general, the physical observables must be symmetric under all crystal symmetries. Example: In general, the conductivity σ is given by the linear response relation j α = α σ αα E α, α,α = x, y, z. 8 (j and E are current and electric field, respectively). The matrix σ must be invariant under the point-group operations of the crystal, DσD 1! = σ.

9 For cubic symmetry, this implies already that the matrix σ is a scalar (proportional to the 1 identity matrix): Rotations by 180 around the z axis, with D = 1, imply 1 σ 11 σ 12 σ 13 σ 11 σ 12 σ 13 σ 11 σ 12 σ 13 D σ 21 σ 22 σ 23 D 1 = σ 21 σ 22 σ 23 =! σ 21 σ 22 σ 23, σ 31 σ 32 σ 33 σ 31 σ 32 σ 33 σ 31 σ 32 σ 33 hence σ 12 = σ 13 = σ 21 = σ 31 =0. Similar rotations around the y and z axis require the matrix to be diagonal. Rotations by 120 around the body-diagonal (which permute x,y,z) imply that σ 11 = σ 22 = σ 33. The reziprocal lattice Definition: For a Bravais lattice G = { R}, the reziprocal lattice is defined by the points G = { G G R =2πn with n Z for all R G}. (6) The reziprocal lattice is a Bravais lattice itself, with primitive vectors a 2 a 3 b1 =2π a 1 (a 2 a 3 ), a 3 a 1 b2 =2π a 1 (a 2 a 3 ), a 1 a 2 b3 =2π a 1 (a 2 a 3 ). Note: The reziprocal lattice need not be of the same lattice type (fcc bcc). Example: If a i are orthogonal, a 1 = a x ê x, a 2 = a y ê y, a 3 = a z ê z, then b 1 = 2π a x ê x, b 2 = 2π a y ê y, b 3 = 2π a z ê z. a 2 lattice reziprocal lattice a 1 b 2 = 2π a 2 b 1 = 2π a 1 Figure 4: Reziprocal lattice of an orthogonal lattice in dimensions d =2. The reziprocal lattice is important in three aspects, which are described below: 9

10 1) Fourier-transformation of space periodic functions If f(r ) is a function which has the lattice periodicity, i.e., f(r + R)=f(r ) for all R G, then the only non-zero Fourier coefficients of f appear at the points of the reziprocal lattice. In short, f k = d 3 rf(r )e ikr = d 3 rf(r + R)e ikr = d 3 rf(r )e i k(r R) = e i k R f k, (where in the first equality we used translational invariance of f, the second equality is a variable transform), i.e., k must satisfy e i kr =1for! all R G, which is equivalent to k G by the definition (6). In general, the fourier series representation of f and its inverse is given by (V is the volume of the unit cell) f(r )= f G e i Gr G G f G = 1 d 3 rf(r )e i Gr V unit-cell (7) (8) 10

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